# Properties

 Label 32.32.7116665922...1057.1 Degree $32$ Signature $[32, 0]$ Discriminant $193^{31}$ Root discriminant $163.73$ Ramified prime $193$ Class number $1$ (GRH) Class group Trivial (GRH) Galois group $C_{32}$ (as 32T33)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14969, 2830728, 59339724, -214409519, -382096988, 2262714296, -1290174274, -6566502528, 11610249674, -638400055, -15151938680, 12702259726, 3732760107, -10578598738, 3455889877, 3133974961, -2529499095, -77972047, 681732194, -170361963, -84476060, 44434017, 2808135, -5339623, 485064, 350581, -65768, -12479, 3538, 210, -93, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 93*x^30 + 210*x^29 + 3538*x^28 - 12479*x^27 - 65768*x^26 + 350581*x^25 + 485064*x^24 - 5339623*x^23 + 2808135*x^22 + 44434017*x^21 - 84476060*x^20 - 170361963*x^19 + 681732194*x^18 - 77972047*x^17 - 2529499095*x^16 + 3133974961*x^15 + 3455889877*x^14 - 10578598738*x^13 + 3732760107*x^12 + 12702259726*x^11 - 15151938680*x^10 - 638400055*x^9 + 11610249674*x^8 - 6566502528*x^7 - 1290174274*x^6 + 2262714296*x^5 - 382096988*x^4 - 214409519*x^3 + 59339724*x^2 + 2830728*x + 14969)

gp: K = bnfinit(x^32 - x^31 - 93*x^30 + 210*x^29 + 3538*x^28 - 12479*x^27 - 65768*x^26 + 350581*x^25 + 485064*x^24 - 5339623*x^23 + 2808135*x^22 + 44434017*x^21 - 84476060*x^20 - 170361963*x^19 + 681732194*x^18 - 77972047*x^17 - 2529499095*x^16 + 3133974961*x^15 + 3455889877*x^14 - 10578598738*x^13 + 3732760107*x^12 + 12702259726*x^11 - 15151938680*x^10 - 638400055*x^9 + 11610249674*x^8 - 6566502528*x^7 - 1290174274*x^6 + 2262714296*x^5 - 382096988*x^4 - 214409519*x^3 + 59339724*x^2 + 2830728*x + 14969, 1)

## Normalizeddefining polynomial

$$x^{32} - x^{31} - 93 x^{30} + 210 x^{29} + 3538 x^{28} - 12479 x^{27} - 65768 x^{26} + 350581 x^{25} + 485064 x^{24} - 5339623 x^{23} + 2808135 x^{22} + 44434017 x^{21} - 84476060 x^{20} - 170361963 x^{19} + 681732194 x^{18} - 77972047 x^{17} - 2529499095 x^{16} + 3133974961 x^{15} + 3455889877 x^{14} - 10578598738 x^{13} + 3732760107 x^{12} + 12702259726 x^{11} - 15151938680 x^{10} - 638400055 x^{9} + 11610249674 x^{8} - 6566502528 x^{7} - 1290174274 x^{6} + 2262714296 x^{5} - 382096988 x^{4} - 214409519 x^{3} + 59339724 x^{2} + 2830728 x + 14969$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $32$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[32, 0]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$71166659227026865952165739269552697151542080277732014965436610606901057=193^{31}$$ magma: Discriminant(K);  sage: K.disc()  gp: K.disc Root discriminant: $163.73$ magma: Abs(Discriminant(K))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $193$ magma: PrimeDivisors(Discriminant(K));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$193$$ Dirichlet character group: $\lbrace$$\chi_{193}(1,·), \chi_{193}(3,·), \chi_{193}(129,·), \chi_{193}(8,·), \chi_{193}(9,·), \chi_{193}(151,·), \chi_{193}(14,·), \chi_{193}(143,·), \chi_{193}(150,·), \chi_{193}(23,·), \chi_{193}(24,·), \chi_{193}(27,·), \chi_{193}(166,·), \chi_{193}(169,·), \chi_{193}(42,·), \chi_{193}(43,·), \chi_{193}(50,·), \chi_{193}(179,·), \chi_{193}(184,·), \chi_{193}(185,·), \chi_{193}(190,·), \chi_{193}(192,·), \chi_{193}(67,·), \chi_{193}(69,·), \chi_{193}(72,·), \chi_{193}(64,·), \chi_{193}(81,·), \chi_{193}(112,·), \chi_{193}(121,·), \chi_{193}(124,·), \chi_{193}(170,·), \chi_{193}(126,·)$$\rbrace$ This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{109} a^{27} - \frac{27}{109} a^{26} + \frac{6}{109} a^{25} + \frac{29}{109} a^{24} - \frac{18}{109} a^{23} - \frac{7}{109} a^{22} + \frac{30}{109} a^{21} + \frac{7}{109} a^{20} + \frac{47}{109} a^{19} + \frac{40}{109} a^{18} - \frac{27}{109} a^{17} - \frac{44}{109} a^{16} + \frac{20}{109} a^{15} - \frac{21}{109} a^{14} - \frac{38}{109} a^{13} - \frac{15}{109} a^{12} + \frac{54}{109} a^{11} + \frac{39}{109} a^{10} - \frac{42}{109} a^{9} + \frac{16}{109} a^{8} - \frac{51}{109} a^{7} - \frac{16}{109} a^{6} - \frac{44}{109} a^{5} - \frac{42}{109} a^{4} + \frac{4}{109} a^{3} + \frac{44}{109} a^{2} - \frac{46}{109} a - \frac{8}{109}$, $\frac{1}{109} a^{28} + \frac{40}{109} a^{26} - \frac{27}{109} a^{25} + \frac{2}{109} a^{24} + \frac{52}{109} a^{23} - \frac{50}{109} a^{22} + \frac{54}{109} a^{21} + \frac{18}{109} a^{20} + \frac{1}{109} a^{19} - \frac{37}{109} a^{18} - \frac{10}{109} a^{17} + \frac{31}{109} a^{16} - \frac{26}{109} a^{15} + \frac{49}{109} a^{14} + \frac{49}{109} a^{13} - \frac{24}{109} a^{12} - \frac{29}{109} a^{11} + \frac{30}{109} a^{10} - \frac{28}{109} a^{9} + \frac{54}{109} a^{8} + \frac{24}{109} a^{7} - \frac{40}{109} a^{6} - \frac{31}{109} a^{5} - \frac{40}{109} a^{4} + \frac{43}{109} a^{3} + \frac{52}{109} a^{2} - \frac{51}{109} a + \frac{2}{109}$, $\frac{1}{109} a^{29} - \frac{37}{109} a^{26} - \frac{20}{109} a^{25} - \frac{18}{109} a^{24} + \frac{16}{109} a^{23} + \frac{7}{109} a^{22} + \frac{17}{109} a^{21} + \frac{48}{109} a^{20} + \frac{45}{109} a^{19} + \frac{25}{109} a^{18} + \frac{21}{109} a^{17} - \frac{10}{109} a^{16} + \frac{12}{109} a^{15} + \frac{17}{109} a^{14} - \frac{30}{109} a^{13} + \frac{26}{109} a^{12} + \frac{50}{109} a^{11} + \frac{47}{109} a^{10} - \frac{10}{109} a^{9} + \frac{38}{109} a^{8} + \frac{38}{109} a^{7} - \frac{45}{109} a^{6} - \frac{24}{109} a^{5} - \frac{21}{109} a^{4} + \frac{1}{109} a^{3} + \frac{42}{109} a^{2} - \frac{11}{109} a - \frac{7}{109}$, $\frac{1}{30193} a^{30} - \frac{21}{30193} a^{29} + \frac{119}{30193} a^{28} - \frac{18}{30193} a^{27} - \frac{5896}{30193} a^{26} + \frac{5587}{30193} a^{25} - \frac{2632}{30193} a^{24} + \frac{11185}{30193} a^{23} - \frac{112}{277} a^{22} - \frac{14132}{30193} a^{21} + \frac{6108}{30193} a^{20} - \frac{13751}{30193} a^{19} - \frac{13957}{30193} a^{18} + \frac{14196}{30193} a^{17} + \frac{14520}{30193} a^{16} + \frac{8387}{30193} a^{15} - \frac{5637}{30193} a^{14} - \frac{11021}{30193} a^{13} + \frac{5628}{30193} a^{12} + \frac{3657}{30193} a^{11} + \frac{13887}{30193} a^{10} + \frac{12904}{30193} a^{9} + \frac{7169}{30193} a^{8} + \frac{10636}{30193} a^{7} + \frac{3487}{30193} a^{6} + \frac{7294}{30193} a^{5} - \frac{11329}{30193} a^{4} + \frac{9574}{30193} a^{3} + \frac{11690}{30193} a^{2} - \frac{14894}{30193} a + \frac{11678}{30193}$, $\frac{1}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{31} + \frac{1396507123818223485209754402329439067571042059663111756146946635332911}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{30} + \frac{306310575189403577215638401094667138072900124115649890840056800156724311}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{29} + \frac{937615056053044181704337557844810989936278489147982403224139671524560}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{28} - \frac{277503096565812892451483024868815493287589885354617601562814781990442912}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{27} + \frac{495594538348463567524052942159185498973226661658161242305486030082952023}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{26} - \frac{29239664252816441021953066167529958674637868800558515142151816100082615774}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{25} + \frac{19272825817189826453104395060327010271851321459367789626269072518190755083}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{24} + \frac{36936754452736965692326848953529318101567517306829608849727557149272150089}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{23} - \frac{38226414025146792408209283112905659797781915964845149794642157930447569318}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{22} - \frac{29992727041974110857442449995723569153249503684502041461517380398734290207}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{21} + \frac{21366213418584076348994202334501387281711043406097336247423776768360160761}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{20} + \frac{6396695472118213122902085878678077058490782344460138652872845153673183806}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{19} - \frac{44755070334935059328611731475611000290664693868147552308895447499243151546}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{18} + \frac{33577852353806166158093536678365248712954497374651669728258948639964971064}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{17} + \frac{14294923369624059370519409177188334632886660499528876373949366638391151457}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{16} + \frac{1493869439959016648845230743977181039916633605135492806121679455320784794}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{15} - \frac{38440885912489752852739586248175533954378040660639438385055119939832054672}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{14} - \frac{8951851251347891676319497780097711487620665729348289021720769878815970781}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{13} - \frac{49685347508415004267472004530813198157508038664639180471881409588657509663}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{12} - \frac{29738303163575678515496353395222995388326315270131652787270538451685611420}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{11} - \frac{16507731964222916824203652210206811201658084033346134938552192538287676055}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{10} + \frac{48188617193349507673316467673641346628173199072753213345100185906536131207}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{9} - \frac{23969766421996308066887458613173998663698757862377092053682136955024895717}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{8} + \frac{49864615462722117696242498717064851454694162570302354556374807762265383202}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{7} + \frac{11011875594612784040558455558604228357922187917154816994113017126236334625}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{6} + \frac{7712480199740460000580286338234456345343292417201590092711384054070504791}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{5} - \frac{14272376344200405912600186440282109343018183866375338551202783773219976668}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{4} + \frac{25718962045426335401757227091941179054489361256365263344696356046912415033}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{3} - \frac{38912817047077705191860450839538794731106976687506448327192392904989014194}{100877246011414201550061355185574630947470476963762602792140977676019421393} a^{2} - \frac{746296672922043619428248489309353485053485191226337700772741401983713917}{100877246011414201550061355185574630947470476963762602792140977676019421393} a - \frac{194799629934785461542633515391544786164568851525441569973020697696165}{6739077160225412622757789777912661563729739926766156910424275347452697}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $31$ magma: UnitRank(K);  sage: UK.rank()  gp: K.fu Torsion generator: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  sage: UK.torsion_generator()  gp: K.tu[2] Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$17081301615656049000000000$$ (assuming GRH) magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A cyclic group of order 32 The 32 conjugacy class representatives for $C_{32}$ Character table for $C_{32}$ is not computed

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $16^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ $32$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ $32$ $32$ $32$ $32$ $16^{2}$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ $32$ $32$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

magma: idealfactors := Factorization(p*Integers(K)); // get the data

magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

gp: idealfactors = idealprimedec(K, p); \\ get the data

gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
193Data not computed